In this video I go over the derivation for the hyperbolic trig identity sinh(x + y) = sinh(x)cosh(y) + cosh(x)sinh(y). This derivation is very similar to the one I did for cosh(x + y) in that I show how we need to think of the final result we want before we start the derivation. In other words we need to use the definition of cosh and sinh by multiplying them together to obtain sinh(x+y). Note though that in my earlier video several years back I derived this same identity but by simply working backwards proving the given identity. But in this video I work from start to finish to show how we sometimes have to think outside of the box instead of the typical linear thinking derivations; so make sure to watch this video!
In this video I go over another example on calculating the area of polar curves and this time find the area enclosed by a circle yet separated by a cardioid. The Cardioid, or Heart in Greek, is the same formula I had previously graphed in my earlier example video. To save time, I plot both of the graphs together by using the Desmos calculator to obtain the resulting area we need to calculate. The first step in solving for the area is to find the points at which they intersect. From there we can apply the formula for the area of a polar curve which I covered in my earlier video, but this time we solve for the area of a polar circle, and then subtract from it the area of the polar cardioid. This is a very important video in understanding how to go about solving in detail the area enclosed by two polar curves, so make sure to watch this video!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvVAKCYLCqdXeyXJow
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/video-notes-polar-coordinates-area-example-2-cardioid-and-circle
Related Videos:
Polar Coordinates: Area: Example 1: r = cos 2ϴ: https://youtu.be/6kavQyGcODo
Polar Coordinates: Area Formula: https://youtu.be/wd-EwiKzqa0
Polar Coordinates: Graphing With Polar Curves with Desmos Calculator: https://youtu.be/xY9Be9HHRsk
Polar Coordinates: Cartesian Connection: https://youtu.be/HcaTYrpmGaU
Polar Coordinates: https://youtu.be/-KAdZL-N4ok
Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw
Polar Coordinates: Example 7: Cardioid: https://youtu.be/rPErcaqNUIY
Half Angle Trigonometry Identities: http://youtu.be/0bY6tHZhBSI .
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https://www.youtube.com/watch?v=vSzxzNn3EiA
In this video I quickly go over a list of common Maclaurin series and their radius of convergence. The first series is for the function 1/(1-x), then e^x, sin x, cos x, arctan x, and finally ln(1+x). Note that in my earlier video I had used an earlier version of my Stewart calculus textbook, which went over the Maclaurin series for ln(1-x) instead of ln(1+x). I quickly recap on that derivation and show that we can simply input -x into ln(1-x) to obtain the Maclaurin series for ln(1+x).
The timestamps of key parts of the video are listed below:
- Question 11: 0:00
- Solution: Recap on Taylor and Maclaurin series: 0:53
- (a) to (f) Table of Maclaurin series: 1:33
- Note on the Maclaurin series for ln(1+x): 5:00
- Maclaurin series for ln(1-x): 5:18
- Replacing x with -x to get ln(1+x): 6:00
- Maclaurin series for ln(1+x): 8:08
This video was taken from my earlier video listed below:
- Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI
- HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L
Related Videos:
Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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This is a short video explaining why the sum of the angles in any triangle is always equal to 180 degrees. You may have taken this fact for granted during high school but the proof to it is quite simple.
Video notes and playlist:
- PDF notes: https://1drv.ms/b/s!As32ynv0LoaIiuNNSnmU8cTiuMn28A?e=rHhCj7
- HIVE notes: https://peakd.com/hive-128780/@mes/angles-in-triangle-equal-180-degrees
- Math proofs: https://www.youtube.com/playlist?list=PL0FC57188710456CF
Related Videos:
Angles - Degrees vs Radians: What are Radians??: http://youtu.be/bOdrVvQWRdE
What is Pi and Proof that it equals 3.14159265....: http://youtu.be/mZc7Uvae4Fw .
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In this video I go over another problems plus example and this time show how to solve for exactly one solution when given two functions with a variable that can be changed around.
Download the notes in my video:
Related Videos:
Problems Plus Example: Double Tangent Lines: http://youtu.be/x2_FjSEFA-Q
Problems Plus Example 1: http://youtu.be/YjoV9TLICOI
Problems Plus - Example 2: Solve hypotenuse in terms of the Perimeter: http://youtu.be/kvtjSQ2g1UE
Problems Plus - Example 3: |x-3| + |x+2| less than 3: http://youtu.be/EdQBtuo2gqM
Problems Plus Example 4 - Mathematical Induction: http://youtu.be/WdIr_onvUtE
Problems Plus Example 5: Introducing a New Variable to Solve Problems: http://youtu.be/UDnH-p6wR6U
Derivative of y = x^n: Power Rule Part 1: n is positive integer: http://youtu.be/-Yv85MZNYgU
Tangent and Secant Lines: http://youtu.be/9E6fBySBsYw
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMPE .
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https://www.youtube.com/watch?v=qC7SGr99ox0
In this video I go over a quick recap on how the sum of a series affects the limit of its terms and its partial sums. If an infinite series sums up to 3, then this must mean that the terms of the series all approach the limit of 0 while the limit of the partial sums approach 3. In other words, the limit of the sequence of partial sums is equal to the sum of the infinite series.
The timestamps of key parts of the video are listed below:
- Question 4: 0:00
- Solution: 0:47
- Limits of the sequence vs partial sums: 1:06
This video was taken from my earlier video listed below:
- Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI
- HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L
Related Videos:
Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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https://www.youtube.com/watch?v=10SkjrJBqbs
In this video I go over the derivative of hyperbolic sine or sinh(x) and show that it equals cosh(x).
Download the notes in my video: https://www.dropbox.com/s/giu7kh35vmlv5tu/438%20-%20Derivative%20Hyperbolic%20Functions%20Proof%20-%20sinh%28x%29.pdf
Related Videos:
Hyperbolic Trigonometry Identity Proof: sinh(-x) = -sinh(x), cosh(-x) = cosh(x): http://youtu.be/-Wqq4Wzi7O8
Hyperbola - Definition and derivation of the equation: x^2/a^2 - y^2/b^2 = 1: http://youtu.be/Y6iYC4VEAi0
Hyperbolic Functions - tanh(x), sinh(x), cosh(x) - Introduction: http://youtu.be/EmJKuQBEdlc
Derivative of Hyperbolic functions - y = sinh(x), tanh(x), cosh(x): http://youtu.be/tdlbLP5pzis
Derivative of Inverse Hyperbolic Functions: inverse sinh(x), cosh(x), tanh(x): http://youtu.be/ubYbYbJOsNs
The Number e - A Brief Introduction and it's Derivative: http://youtu.be/o_s_YYD6v3g
Derivative of y = x^n: Power Rule Part 1: n is positive integer: http://youtu.be/-Yv85MZNYgU .
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https://www.youtube.com/watch?v=9QpIDqYg52c
In my earlier videos I went over a useful strategy for integration but with that a very important question arises: can we use our integration strategy to integrate all continuous functions? The answer to that question is unfortunately NO, well at least not in terms of the regular elementary functions. In this video I go over what elementary functions are and explain how most elementary functions, such as e^(x^2) can not be integrated in terms of basic elementary functions. One method to prove whether or not a function can be integrated in terms of elementary functions is called the Risch Algorithm and was developed in 1968 by the American Mathematician Robert Henry Risch.
In my later videos I will go over how functions such as e^(x^2) can be integrated in terms of infinite series so stay tuned for that!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhd4iIg7HrqROZAvF2w
Related Videos:
Strategy for Integration: Examples 2 to 5: http://youtu.be/3JGU3vJsPIk
Strategy for Integration: Example 1: http://youtu.be/xqwRpvdrD6Y
Strategy for Integration: http://youtu.be/7JxW2N7izF0
Integration by Partial Fractions: http://youtu.be/r07NnKf76og
Trigonometric Substitution for Integrals: http://youtu.be/2pWvGXwtVJo
Integration by Parts: Proof: http://youtu.be/TZhEOct5u_M
The Substitution Rule for Integrals: http://youtu.be/VsLC-0g6hVg
Trigonometric Integrals: Guidelines for sin(x)cos(x): http://youtu.be/aE8Sz5BdMT8
Trigonometric Integrals: Guidelines for tan(x)cos(x): http://youtu.be/oz5nXlYQPB0 .
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https://www.youtube.com/watch?v=OFEDLJYqYps
In this video my brother Mark aka MFA goes over how the NBA's new in-season tournament works and how each group is selected. The tournament involves splitting each conference into 3 groups of 5 teams, which are randomly drawn from 5 pots of 3 teams each (listed in order of last year's season rankings). The group stage involves playing each team inside the group twice, one road and one away, while the games count towards the regular season. The top team in each group, and 1 wildcard from each conference make it to the knockout stage. The first knockout stage is the quarter finals, while the losing teams play each other in each conference; once again all games here count towards the regular season. Next is the Semi-finals, which also count towards the regular season. The Finals will be held in Las Vegas and will not count towards the regular season. Every player in the knockout stages gets a financial bonus, while the champion gets $500,000 per player! Note also that every in-season tournament game will have the court painted in custom tournament colors.
This is the first year that the National Basketball Association (NBA) has included a tournament within the regular season. The goal is to increase the excitement around the early part of the long NBA season. I think it's a great idea, what do you think?
The timestamps of key parts of the video are listed below:
- NBA In-Season Tournament: 0:00
- Group Play: 0:36
- Knockout Rounds: Quarter Finals: 1:39
- Semifinals: 2:25
- Finals: 2:44
- Conclusion: 3:12
- Tie Breaker: 3:51
- Choosing the Groups: 4:16
- Example: Eastern Conference: 4:49
- What do you think of the Tournament? 5:37
Download Video Notes:
- PDF: https://1drv.ms/b/s!As32ynv0LoaIirpTLmaguLb8DrxC0g
- HIVE: https://peakd.com/nba/@mes/nba-in-season-tournament-how-does-it-work
Related Videos:
NBA basketball tutorials: https://www.youtube.com/playlist?list=PL43B7E7AE41C847FA .
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https://www.youtube.com/watch?v=nVXJmgburhw
In this video I recap on Taylor polynomials, Taylor series, Maclaurin series, and Taylor's inequality. A Taylor polynomial is a finite sum with terms consisting of the i-th derivative at a divided by the i-th factorial and multiplied by (x - a) to the power of i. A Taylor series is just an infinite Taylor polynomial. A Maclaurin series is a special type of Taylor series where a = 0. A function is equal to its Taylor series if the remainder approaches 0 as n approaches infinity. Taylor's Inequality states that if the (n + 1) derivative of f(x) is less than or equal to a number M then the absolute value of the n-th remainder of a Taylor series is less than M divided by (n + 1) factorial multiplied by the absolute value of (x - a) to the power of (n + 1). Taylor and Maclaurin series are very useful since they can turn complicated functions into a simple sum of many easily calculatable terms, and is how most calculators work behind the scenes!
The timestamps of key parts of the video are listed below:
- Question 10: 0:00
- (a) Taylor polynomial: 0:42
- (b) Taylor series: 2:20
- (c) Maclaurin series: 3:33
- (d) A function equals its Taylor series if the remainder approaches 0: 4:26
- (e) Taylor's Inequality: 5:45
This video was taken from my earlier video listed below:
- Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI
- HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L
Related Videos:
Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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